Related papers: Primitive quantales

In order to consider j-wise relative r-primality conditions that do not necessarily require all j-tuples of elements in a Dedekind domain to be relatively r-prime, we define the notion of j-wise relative r-primality with respect to a fixed…

Let R be a commutative ring with identity and N(R) be the set of all nilpotent elements of R. The aim of this paper is to introduce and study the notion of nil-prime ideals as a generalization of prime ideals. We say that a proper ideal P…

For an integral domain $R$ satisfying certain condition, we characterize the primitive ideal space and its Jacobson topology of the semigroup crossed product $C^*(R_+) \rtimes R^\times$. The main example is when $R=\mathbb{Z}[\sqrt{-3}]$.

Jacobson's commutativity theorem says that a ring is commutative if, for each $x$, $x^n = x$ for some $n > 1$. Herstein's generalization says that the condition can be weakened to $x^n-x$ being central. In both theorems, $n$ may depend on…

It has been established on many occasions that the set of quotients of prime numbers is dense in the set of positive real numbers. More recently, it has been proved that the set of quotients of primes in the Gaussian integers is dense in…

Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements, and let $m_1$ and $m_2$ be positive integers. Given polynomials $f_1(x), f_2(x) \in \mathbb{F}_q[x]$ with $\textrm{deg}(f_i(x)) \leq m_i$, for $i = 1, 2$, and such that the…

A theorem of Cohen from 1950 states that a commutative ring is Noetherian if and only if every prime ideal is finitely generated. In this note, we establish analogues of this result in tensor triangular geometry. In particular, for an…

In this note we classify the primitive ideals in finite W-algebras of type A.

We establish primitive recursive versions of some known facts about computable ordered fields of reals and computable reals, and then apply them to proving primitive recursiveness of some natural problems in linear algebra and analysis. In…

We show that a generalization of the theory of quantales and prequantales provides a noncommutative and nonassociative abstract ideal theoretic setting for the theories of star operations, semistar operations, semiprime operations, ideal…

We extend the classical notion of standardly stratified $k$-algebra (stated for finite dimensional $k$-algebras) to the more general class of rings, possibly without $1,$ with enough idempotents. We show that many of the fundamental…

The general Nullstellensatz states that if $A$ is a Jacobson ring, $A[X]$ is Jacobson. We introduce the notion of an $\alpha$-Jacobson ring for an ordinal $\alpha$ and prove a quantitative version of the general Nullstellensatz: if $A$ is…

This is a survey of what is known and/or conjectured about the prime and primitive spectra of quantum algebras, of quantized coordinate rings in particular. The topological structure of these spectra, their relations to classical affine…

For $q=3^r$ ($r>0$), denote by $\mathbb{F}_q$ the finite field of order $q$ and for a positive integer $m\geq2$, let $\mathbb{F}_{q^m}$ be its extension field of degree $m$. We establish a sufficient condition for existence of a primitive…

This paper studies the multiplicative ideal structure of commutative rings in which every finitely generated ideal is quasi-projective. Section 2 provides some preliminaries on quasi-projective modules over commutative rings. Section 3…

Completely prime right ideals are introduced as a one-sided generalization of the concept of a prime ideal in a commutative ring. Some of their basic properties are investigated, pointing out both similarities and differences between these…

A complete understanding of the structure of all prime ideals of an infinite direct product of commutative rings (e.g. in terms of more specific objects) has remained a challenging problem for decades. In this article, new advances have…

Let $G$ be a finite group and $p$ be a prime. We study the kernel of the map, between the Burnside ring of $G$ and the Grothendieck ring of $\mathbb{F}_p[G]$-modules, taking a $G$-set to its associated permutation module. We are able, for…

We investigate prime avoidance for an arbitrary set of prime ideals in a commutative ring. Various necessary and/or sufficient conditions for prime avoidance are given, which yield natural classes of infinite sets of primes that satisfy…

In this paper, we introduce a new generalization of weakly prime ideals called $I$-prime. Suppose $R$ is a commutative ring with identity and $I$ a fixed ideal of $R$. A proper ideal $P$ of $R$ is $I$-prime if for $a, b \in R$ with $ab \in…